What are the best math instructional sites, resources and (especially) tools for graphing and math visualization these days?

It's been over 20 years since my mathematical high-water-mark (college calculus), but I'm trying to get the ol' brain back in shape. I'm working my way through algebra, geometry, and trig once more, with the goal of eventually getting through calculus in a classroom again. Also, I'm a homeschool parent and my kids aren't that far behind, so this question might help me teach them, too.

Back in the day, we had non-graphing calculators and had to do all of our visualization on graph paper or mentally. Outside of class time, our resources were limited to the textbook and the teacher's office hours. I hear a lot's changed -- something about "computers" and an "information superhighway"? What are some good online & software tools for helping me get it as I pursue these goals?

I love Vi Hart, but she's more dessert than main course. I've dipped my toe into the waters of Khan Academy and found it helpful, but it seems to lack the organizational layer a textbook would have. I've heard of Wolfram Alpha but don't really know where to start. Tips on getting the most out of these resources are welcome, as are any stuff I haven't heard of. Free preferred, but excellent paid resources welcome too.

For the record, my primary computer is running Linux, but I have access to machines running Win7 and MacOSX, too.

Jan Gullberg's _Mathematics: From the Birth of Numbers_ is a good overview of many different subsets of math, what they're good for, and so on. It includes historical context as well; that may help it to feel more cohesive.

I have a bunch of math-related resources I could forward along if you're looking for something more specific, and especially if you have any programming experience.posted by silentbicycle at 9:44 AM on August 24, 2011

Did you look at the examples pages on Wolfram Alpha? The examples are really comprehensive and, for me, do a good job in terms of inspiration.posted by anaelith at 9:45 AM on August 24, 2011

As for software resources, J has a many excellent tutorials ("labs") included with the standard installation. J is a bit unconventional, as math-centric programming languages go, but quite powerful.

Silentbike, I am a PHP/SQL programmer by trade, so programming resources are welcome.posted by richyoung at 9:51 AM on August 24, 2011

Yea, I just got the trial of Mathematica from Wolfram Alpha a few days ago, and it's incredibly awesome so far. They have some videos and examples to get you started.posted by amsterdam63 at 9:54 AM on August 24, 2011

richyoung: Project Euler! Once you solve the problems, you can check out other peoples' solutions, in a variety of languages. I learned quite a bit about algorithms and optimization by doing problems in OCaml, C, and Lua. It's also a good math refresher.

Also, if you have other programming experience, J will be extra mind-bending. It's a modern APL dialect, which puts it quite far from most conventional languages.

Gilbert Strang's opencourseware lectures on Linear Algebra are quite good. He has a good, free calculus textbook as well.

Also, check out Knuth et. al.'s _Concrete Mathematics_.

This question comes up on Hacker News periodically. Here are a couple threads there - (1), (2), (3).posted by silentbicycle at 10:08 AM on August 24, 2011

It may be too basic for you, but I kind of love using Paul's Online Math Notes as a refresher when I need to revisit something I've learned and forgotten. I tend to like doing math the old-fashioned way, though, by sitting down with a pencil and a notebook and working my way through a bunch of example problems.posted by pullayup at 10:25 AM on August 24, 2011

I recommend using the student edition of Matlab (which allows you to do numerical computations, create plots, wrangle data & do statistics, and do symbolic math). There is an open source project that aims to replicate the functionality of Matlab called GNU Octave, but in my experience doesn't replicate the same ease of use of Matlab (kind of jury-rigged and the syntax isn't completely the same as Matlab).

I would add that while all these graphing tools and computer-aided algebra systems are all fun and stuff, they don't replace old-fashioned hand sketching of functions, geometric constructions with ruler and compass, hand derivations of derivatives and integrals. There's something about doing math by hand without the use of fancy tools that helps with solidifying and generalizing the knowledge and concepts learned.posted by scalespace at 11:22 AM on August 24, 2011 [1 favorite]

This summer I've become enamoured with mathcentre. They've got an algebra refresher booklet and a calculus refresher booklet, as well as stuff broken down by topic. It's designed to be a quick reference site, so it's maybe a little hard to navigate for your purposes. The University of Plymouth has a whole curriculum online, as well.

I'm not entirely sure what you're looking for software-wise, as I'm not sure what's out there that would give you a big advantage over a graphing calculator. WolframAlpha probably comes the closest. In terms of free stuff, you're looking at Octave, that scalespace mentioned, and sage. While I have taught a class that used Matlab (and used Octave at home), I don't really do numerical stuff, so I don't have complaints about Octave. I didn't even know Matlab could do symbolic manipulation (apparently it's calling Maple somehow) and would suggest sage is really want you want for this sort of thing. (There's certainly nothing wrong with Maple or Mathematica either, except they cost money.)posted by hoyland at 12:50 PM on August 24, 2011

Thanks, everyone. I think this will keep me busy for a while, especially following up on the Hacker News links. I was looking for visualization tools and instructional materials, so things like gnuplot or Matlab/Octave fit the former nicely, while the opencourseware link, mathcentre and Paul's notes fit the latter. But I like the Project Euler format as well, and may challenge my son (who fancies himself a python programmer) to a friendly competition there.

I've installed J, but I'm not sure I'm willing to smoke whatever it is one must smoke to make sense of it.... Wolfram does some visualization, but every now & then, it seems to just solve the problem for me instead of plotting it, which I don't like. Installed a GUI front end to gnuplot, which made sense quickly; installed but haven't looked at Octave yet, and R is installing as I type this, so we'll see how that goes. (And it goes without saying that I will keep a sharp pencil & graph paper handy as scalespace recommends.) Thanks again to all who responded.posted by richyoung at 9:47 AM on August 25, 2011

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